Strict Single Crossing and the Strict Spence-Mirrlees Condition: A Comment on Monotone Comparative Statics - Edlin, Aaron S.; Shannon, Chris (Econometrica, 1998)
MILGROM AND SHANNON (1994) clarify the relationship between order-theoretic methods for comparative statics and more traditional differential techniques by developing relationships between the differential Spence-Mirrlees single crossing property and the order-theoretic single crossing property. Both conditions are central for monotone comparative statics analysis in a number of settings. In particular, Milgrom and Shannon show that the order-theoretic single crossing property is necessary and sufficient for the set of optimal choices to be nondecreasing in certain choice problems, and that a strict form of the single crossing property guarantees the stronger conclusion that every selection from the set of maximizers is nondecreasing in such problems. Milgrom and Shannon assert that under appropriate conditions the Spence-Mirrlees condition is equivalent to their single crossing property, and that the strict versions are also equivalent. In this note, however, we give counterexamples which show that their strict single crossing property may hold even though the strict Spence-Mirrlees condition fails. In fact, we show that the strict single crossing property may hold even though the strict Spence-Mirrlees condition holds only on a set of arbitrarily small measure. We also give a correct statement of the relationship between the Spence-Mirrlees condition and the single crossing property. These counterexamples explain the discrepancy between the monotonicity conclusions that Milgrom and Shannon (1994) derive from the strict single crossing property and the strict monotonicity conclusions that Edlin and Shannon (1998) derive from the strict Spence-Mirrlees condition. In Section 3 we also use these counterexamples to illustrate the fact that the strict single crossing property can allow both pooling and separating equilibria while the strict Spence-Mirrlees condition eliminates the possibility of pooling equilibria. The elimination of pooling equilibria in signalling and screening models is more subtle than Edlin and Shannon's (1998) strict monotonicity conclusions because agents need not face a differentiable constraint.